We give an introduction to the theory of multi-partite entanglement. We begin
by describing the "coordinate system" of the field: Are we dealing with pure or
mixed states, with single or multiple copies, what notion of "locality" is
being used, do we aim to classify states according to their "type of
entanglement" or to quantify it? Building on the general theory of
multi-partite entanglement - to the extent that it has been achieved - we turn
to explaining important classes of multi-partite entangled states, including
matrix product states, stabilizer and graph states, bosonic and fermionic
Gaussian states, addressing applications in condensed matter theory. We end
with a brief discussion of various applications that rely on multi-partite
entangled states: quantum networks, measurement-based quantum computing,
non-locality, and quantum metrology.

In short-range interacting systems, the speed at which entanglement can be
established between two separated points is limited by a constant Lieb-Robinson
velocity. Long-range interacting systems are capable of faster entanglement
generation, but the degree of the speed-up possible is an open question. In
this paper, we present a protocol capable of transferring a quantum state
across a distance $L$ in $d$ dimensions using long-range interactions with
strength bounded by $1/r^\alpha$. If $\alpha < d$, the state transfer time is
asymptotically independent of $L$; if $\alpha = d$, the time is logarithmic in
distance $L$; if $d < \alpha < d+1$, transfer occurs in time proportional to
$L^{\alpha - d}$; and if $\alpha \geq d + 1$, it occurs in time proportional to
$L$. We then use this protocol to upper bound the time required to create a
state specified by a MERA (multiscale entanglement renormalization ansatz)
tensor network, and show that, if the linear size of the MERA state is $L$,
then it can be created in time that scales with $L$ identically to state
transfer up to multiplicative logarithmic corrections.

We analyze the equivalence between discrete-time coined quantum walks and
Szegedy's quantum walks. We characterize a class of flip-flop coined models
with generalized Grover coin on a graph $\Gamma$ that can be directly converted
into Szegedy's model on the subdivision graph of $\Gamma$ and we describe a
method to convert one model into the other. This method improves previous
results in literature that need to use the staggered model and the concept of
line graph, which are avoided here.

Transport phenomena in parallel coupled scatterers are studied by transfer
matrix formulism. We derive a simple recurrence relation for transfer matrix of
one-dimensional two-terminal systems consisting of $N$ arbitrary scattering
unit cells connected in parallel. For identical scattering sub-units we find
that the effects of parallel connection on transport properties of the coupled
system can be described by a similarity transformation on the single scatterer,
with the similar matrix determined by the scattering matrix of the junction.
While for distinct single scatterers, the similar matrices depend on both
scattering properties of individual elements and structure of connection
topologies.

We study the scattering properties of $N$ identical one-dimensional localized
$\mathcal{PT}$-symmetric potentials, connected in series as well as in
parallel. We derive a general transfer matrix formalism for parallel coupled
quantum scatterers, and apply that theory to demonstrate that the spectral
singularities and $\mathcal{PT}$-symmetric transitions of single scattering
cells may be observed in coupled systems, at the same or distinct values of the
critical parameters, depending on the connection modes under which the
scattering objects are coupled. We analyse the influences of the connection
configuration on the related transport properties such as spectral
singularities and anisotropic transmission resonances.

Comments: 24 pages, 14 figures. Contribution to "Lectures on general quantum correlations and their applications", edited by Felipe Fanchini, Diogo Soares-Pinto, and Gerardo Adesso. Sorry for being late!

Subjects:Quantum Physics (quant-ph)

In this chapter we review the contributions of Nuclear Magnetic Resonance to
the study of quantum correlations, including its capabilities to prepare
initial states, generate unitary transformations, and characterize the final
state. These are the three main demands to implement quantum information
processing in a physical system, which NMR offers, nearly to perfection, though
for a small number of qubits. Our main discussion will concern liquid samples
at room temperature.

Complementarity and nonlocality, these two traits are believed to differ
quantum physics from the classical physics. In this paper, we introduce a
quantity and prove many people's belief that the complementarity between global
observables and local observables set the decisive foundation for the
nonlocality of composite systems, by focusing on the maximal extent of these
two abilities for a composite system.

The dynamical Casimir effect (DCE) is the production of photons by the
amplification of vacuum fluctuations. In this paper we demonstrate new
resonance conditions in DCE that potentially allow the production of optical
photons when the mechanical frequency is smaller than the lowest frequency of
the cavity field. We consider a cavity with one mirror fixed and the other
allowed to oscillate. In order to identify the region where production of
photons takes place, we do a linear stability analysis and investigate the
dynamic stability of the system under small fluctuations. By using a numerical
solution of the Heisenberg equations of motion, the time evolution of the
number of photons produced in the unstable region is studied.

We investigate entanglement and coherence in an $XXZ$ spin-$s$ pair immersed
in a non-uniform transverse magnetic field. The ground state and thermal
entanglement phase diagrams are analyzed in detail in both the ferromagnetic
and antiferromagnetic cases. It is shown that a non-uniform field enables to
control the energy levels and the entanglement of the corresponding
eigenstates, making it possible to entangle the system for any value of the
exchange couplings, both at zero and finite temperatures. Moreover, the limit
temperature for entanglement is shown to depend only on the difference
$|h_1-h_2|$ between the fields applied at each spin, leading for $T>0$ to a
separability stripe in the $(h_1,h_2)$ field plane such that the system becomes
entangled above a threshold value of $|h_1-h_2|$. These results are
demonstrated to be rigorously valid for any spin $s$. On the other hand, the
relative entropy of coherence in the standard basis, which coincides with the
ground state entanglement entropy at $T=0$ for any $s$, becomes non-zero for
any value of the fields at $T>0$, decreasing uniformly for sufficiently high
$T$. A special critical point arising at $T=0$ for non-uniform fields in the
ferromagnetic case is also discussed

Orbital angular momentum of photons is an intriguing system for the storage
and transmission of quantum information, but it is rapidly degraded by
atmospheric turbulence. We explore the ability of adaptive optics to compensate
for this disturbance by measuring and correcting cumulative phase shifts in the
wavefront. These shifts can be represented as a sum of Zernike functions; we
analyze the residual errors after correcting up to a certain number of Zernike
modes when an orbital angular momentum state is transmitted through a turbulent
atmosphere whose density fluctuations have a Kolmogorov spectrum. We
approximate the superoperator map that represents these residual errors and
find the solution in closed form. We illustrate with numerical examples how
this perturbation depends on the the number of Zernike modes corrected and the
orbital angular momentum state of the light.

We propose a method of trapping atoms in arrays near to the surface of a
composite nanophotonic device with optimal coupling to a single cavity mode.
The device, comprised of a nanofiber mounted on a grating, allows the formation
of periodic optical trapping potentials near to the nanofiber surface along
with a high cooperativity nanofiber cavity. We model the device analytically
and find good agreement with numerical simulations. We numerically demonstrate
that for an experimentally realistic device, an array of traps can be formed
whose centers coincide with the antinodes of a single cavity mode, guaranteeing
optimal coupling to the cavity. Additionally, we simulate a trap suitable for a
single atom within 100 nm of the fiber surface, potentially allowing larger
coupling to the nanofiber than found using typical guided mode trapping
techniques.

Measurement outcomes of a quantum state can be genuinely random
(unpredictable) according to the basic laws of quantum mechanics. The
Heisenberg-Robertson uncertainty relation puts constrains on the accuracy of
two non-commuting observables. The existing uncertainty relations adopt
variance or entropic measures, which are functions of observed outcome
distributions, to quantify the uncertainty. According to recent studies of
quantum coherence, such uncertainty measures contain both classical
(predictable) and quantum (unpredictable) components. In order to extract out
the quantum effects, we define quantum uncertainty to be the coherence of the
state on the measurement basis. We discover a quantum uncertainty relation of
coherence between two measurement non-commuting bases. Furthermore, we
analytically derive the quantum uncertainty relation for the qubit case with
three widely adopted coherence measures, the relative entropy of coherence, the
coherence of formation, and the $l_1$ norm of coherence.

The dynamics of tripartite entanglement of fermionic system in noninertial
frames through linear contraction criterion when one or two observers are
accelerated is investigated. In one observer accelerated case the entanglement
measurement is not invariant with respect to the partial realignment of
different subsystems and for two observers accelerated case it is invariant. It
is shown that the acceleration of the frame does not generate entanglement in
any bipartite subsystems. Unlike the bipartite states, the genuine tripartite
entanglement does not completely vanish in both one observer accelerated and
two observers accelerated cases even in the limit of infinite acceleration. The
degradation of tripartite entanglement is fast when two observers are
accelerated than when one observer is accelerated. It is shown that tripartite
entanglement is a better resource for quantum information processing than the
bipartite entanglement in noninertial frames .

The security of quantum key distribution protocols hinge upon some uniquly
quantum features of the physical systems. We explore the role of contextuality
in this context and present a quantum key distribution protocol based on
quantum contextuality. The KCBS inequality and contextuality monogamy is
exploited to show that our protocol is secure. We explicitly calculate the key
rate, error rate introduced between Alice-Bob communication due to the presence
of Eve and the information gain by Eve. This protocol provides a new framework
for quantum key distribution which has conceptual and practical advantages over
other quantum protocols.

The non-Markovian nature of quantum systems recently turned to be a key
subject for investigations on open quantum system dynamics. Many studies, from
its theoretical grounding to its usefulness as a resource for quantum
information processing and experimental demonstrations, have been reported in
the literature. Typically, in these studies, a structured reservoir is required
to make non-Markovian dynamics to emerge. Here, we investigate the dynamics of
a qubit interacting with a bosonic bath and under the injection of a classical
stochastic colored noise. A canonical Lindblad-like master equation for the
system is derived, using the stochastic wavefunction formalism. Then, the
non-Markovianity of the evolution is witnessed using the Andersson, Cresser,
Hall and Li measure. We evaluate the measure for three different noises and
study the interplay between environment and noise pump necessary to generate
quantum non-Markovianity, as well as the energy balance of the system. Finally,
we discuss the possibility to experimentally implement the proposed model.

We present a new, three-dimensional, topological model for quantum
information. Our picture combines charged excitations carried by strings, with
topological properties that arise from embedding the strings in a three
manifold. A quon is a neutral pair of strings. The properties of the manifold
allow us to interpret multi-quons and their transformations in a natural way.
We obtain a new type of relation, a string-genus "joint relation," between (1):
a diagram that contains a neutral string surrounding the genus of the manifold,
and (2): the diagram without the string and genus. We use the joint relation to
obtain a 3D topological interpretation of the $C^{*}$ Hopf algebra relations,
that are widely used in tensor networks. We obtain a 3D representation of the
CNOT gate and a 3D topological protocol for teleportation.

We study the optimization problem for remote one- and two-qubit state
creation via a homogeneous spin-1/2 communication line using the local unitary
transformations of the multi-qubit sender and extended receiver. We show that
the maximal length of a communication line used for the needed state creation
(the critical length) increases with an increase in the dimensionality of the
sender and extended receiver. The model with the sender and extended receiver
consisting of up to 10 nodes is used for the one-qubit state creation and we
consider two particular states: the almost pure state and the maximally mixed
one. Regarding the two-qubit state creation, we numerically study the
dependence of the critical length on a particular triad of independent
eigenvalues to be created, the model with four-qubit sender without an extended
receiver is used for this purpose.

A spin-$j$ state can be represented by a symmetric tensor of order $N=2j$ and
dimension $4$. Here, $j$ can be a positive integer, which corresponds to a
boson; $j$ can also be a positive half-integer, which corresponds to a fermion.
In this paper, we introduce regularly decomposable tensors and show that a
spin-$j$ state is classical if and only if its representing tensor is a
regularly decomposable tensor. In the even-order case, a regularly decomposable
tensor is a completely decomposable tensor but not vice versa; a completely
decomposable tensors is a sum-of-squares (SOS) tensor but not vice versa; an
SOS tensor is a positive semi-definite (PSD) tensor but not vice versa. In the
odd-order case, the first row tensor of a regularly decomposable tensor is
regularly decomposable and its other row tensors are induced by the regular
decomposition of its first row tensor. We also show that complete
decomposability and regular decomposability are invariant under orthogonal
transformations, and that the completely decomposable tensor cone and the
regularly decomposable tensor cone are closed convex cones. Furthermore, in the
even-order case, the completely decomposable tensor cone and the PSD tensor
cone are dual to each other. The Hadamard product of two completely
decomposable tensors is still a completely decomposable tensor. Since one may
apply the positive semi-definite programming algorithm to detect whether a
symmetric tensor is an SOS tensor or not, this gives a checkable necessary
condition for classicality of a spin-$j$ state. Further research issues on
regularly decomposable tensors are also raised.

Approximate analytical solutions of a two-term potential are studied for the
relativistic wave equations, namely, for the Klein-Gordon and Dirac equations.
The results are obtained by solving of a Riemann-type equation whose solution
can be written in terms of hypergeometric function $\,_{2}F_{1}(a,b;c;z)$. The
energy eigenvalue equations and the corresponding normalized wave functions are
given both for two wave equations. The results for some special cases including
the Manning-Rosen potential, the Hulth\'{e}n potential and the Coulomb
potential are also discussed by setting the parameters as required.

For years, the biggest unspeakable in quantum theory has been why quantum
theory and what is quantum theory telling us about the world. Recent efforts
are unveiling a surprisingly simple answer. Here we show that two
characteristic limits of quantum theory, the maximum violations of
Clauser-Horne-Shimony-Holt and Klyachko-Can-Binicio\u{g}lu-Shumovsky
inequalities, are enforced by a simple principle. The effectiveness of this
principle suggests that non-realism is the key that explains why quantum
theory.

In this paper we investigate system identification for general quantum linear
systems. We consider the situation where the input field is prepared as
stationary (squeezed) quantum noise. In this regime the output field is
characterised by the power spectrum. We address the following two questions:
(1) Which parameters can be identified from the power spectrum? (2) How to
construct a system realisation from the power spectrum? The power spectrum
depends on the system parameters via the transfer function, here we show that
this map is injective under global minimality.

Fault-tolerant quantum computers compose elements of a discrete gate set in
order to approximate a target unitary. The problem of minimising the number of
gates is known as gate-synthesis. The approximation error is a form of coherent
noise, which can be significantly more damaging than comparable incoherent
noise. We show how mixing over different gate sequences can convert this
coherent noise into an incoherent form. As measured by diamond distance, the
post-mixing noise is quadratically smaller than before mixing, with no
additional resource cost. Equivalently, we can use a shorter gate sequence to
achieve the same precision as unitary gate-synthesis, with a factor 1/2
reduction for a broad class of problems.

This letter reports the influence of noisy channels on JRSP of two-qubit
equatorial state. We present a scheme for JRSP of two-qubit equatorial state.
We employ two tripartite Greenberger-Horne-Zeilinger (GHZ) entangled states as
the quantum channel linking the parties. We find the success probability to be
$1/4$. However, this probability can be ameliorated to $3/4$ if the state
preparers assist by transmitting individual partial information through
classical channel to the receiver non-contemporaneously. Afterward, we
investigate the effects of five quantum noises: the bit-flip noise, bit-phase
flip noise, amplitude-damping noise, phase-damping noise and depolarizing noise
on the JRSP process. We obtain the analytical derivation of the fidelities
corresponding to each quantum noisy channel, which is a measure of information
loss as the qubits are being distributed in these quantum channels. We find
that the system loses some of its properties as a consequence of unwanted
interactions with environment. For instance, within the domain
$0<\lambda<0.65$, the information lost via transmission of qubits in amplitude
channel is most minimal, while for $0.65<\lambda\leq1$, the information lost in
phase flip channel becomes the most minimal. Also, for any given $\lambda$, the
information transmitted through depolarizing channel has the least chance of
success.

We study, further, a conjectured formula for generalized two-qubit
Hilbert-Schmidt separability probabilities that has recently been proven by
Lovas and Andai (https://arxiv.org/pdf/1610.01410.pdf) for its real (two-rebit)
asserted value ($\frac{29}{64}$), and that has also been very strongly
supported numerically for its complex ($\frac{8}{33}$), and quaternionic
($\frac{26}{323}$) counterparts. Now, we seek to test the presumptive
octonionic value of $\frac{44482}{4091349} \approx 0.0108722$. We are somewhat
encouraged by certain numerical computations, indicating that this
(51-dimensional) instance of the conjecture might be fulfilled by setting a
certain determinantal-power parameter $a$, introduced by Forrester
(https://arxiv.org/pdf/1610.08081.pdf), to 0 (or possibly near to 0).
Hilbert-Schmidt measure being the case $k=0$ of random induced measure, for
$k=1$, the corresponding octonionic separability probability conjecture is
$\frac{7612846}{293213345} \approx 0.0259635$, while for $k=2$, it is
$\frac{4893392}{95041567} \approx 0.0514869, \ldots$. The relation between the
parameters $a$ and $k$ is explored.

Classical autoencoders are neural networks that can learn efficient codings
of large datasets. The task of an autoencoder is, given an input $x$, to simply
reproduce $x$ at the output with the smallest possible error. For one class of
autoencoders, the structure of the underlying network forces the autoencoder to
represent the data on a smaller number of bits than the input length,
effectively compressing the input. Inspired by this idea, we introduce the
model of a quantum autoencoder to perform similar tasks on quantum data. The
quantum autoencoder is trained to compress a particular dataset, where an
otherwise efficient compression algorithm cannot be employed. The training of
the quantum autoencoder is done using classical optimization algorithms. We
show that efficient quantum autoencoders can be obtained using simple circuit
heuristics. We apply our model to compress molecular wavefunctions with
applications in quantum simulation of chemistry.

We upgrade cMERA to a systematic variational ansatz and develop techniques
for its application to interacting quantum field theories in arbitrary
spacetime dimensions. By establishing a correspondence between the first two
terms in the variational expansion and the Gaussian Effective Potential, we can
exactly solve for a variational approximation to the cMERA entangler. As
examples, we treat scalar $\varphi^4$ theory and the Gross-Neveu model and
extract non-perturbative behavior. We also comment on the connection between
generalized squeezed coherent states and more generic entanglers.

Out-of-time-ordered (OTO) correlation functions have been proposed to
describe the distribution or "scrambling" of information across a quantum
state. In this work, we investigate both time-ordered and OTO correlation
functions in the non-integrable, one-dimensional Bose-Hubbard model at high
temperatures where well-defined quasiparticles cease to exist. Performing
numerical simulations based on matrix product operators, we observe a linear
light-cone spreading of quantum information in the OTO correlators. From our
numerical data, we extract the speed of information propagation and the
Lyapunov exponent, which we compare with predictions from holography. In
contrast with the fast spreading of information, the thermalization of the
system takes parametrically longer due to the slow diffusion of conserved
quantities. Our numerical simulations demonstrate such slow hydrodynamic
power-laws in the late time dynamics of the density correlation function. We
furthermore propose two different interferometric schemes to approach the
challenge of measuring time-ordered as well as OTO correlation functions in
real space and time. Our protocols do not require an ancillary qubit and are
respectively based on the local and global interference of two copies of the
many-body state.

We investigate the prospects of controlling charge-exchange in ultracold
collisions of heteroisotopic combinations of atoms and ions of the same
element. The treatment, readily applicable to alkali or alkanine-earth metals,
is illustrated in the process $^9$Be$^{+}$ + $^{10}$Be $\leftrightarrow$
$^{9}$Be + $^{10}$Be$^{+}$, which exhibits favorable electronic, nuclear, and
hyperfine structure. Feshbach resonances are obtained from quantum scattering
calculations in a standard coupled-channel formalism with non-BO terms
originating from the nuclear kinetic operator. Near a narrow resonance
predicted at 322 G, we find the charge-exchange rate coefficient to rise from
practically zero to values larger than $10^{-12}$ cm$^3$/s. Our results suggest
controllable charge-exchange reactions between different isotopes of suitable
atom-ion pairs with potential applications to quantum systems engineered to
study charge diffusion in trapped cold atom-ion mixtures and emulate many-body
physics.

The Formation of metastable molecules (Feshbach resonances) at the collision
of two atoms and subsequent stimulated transition to a lower unbound electronic
molecular state, with emission of a photon of the laser radiation has been
investigated. This can develop, in particular, for $Rb_2$ molecules due to
resonance scattering of two $Rb$ atoms. The considered process is a basis for
the creation of excimer lasers. Expressions for the cross sections of elastic
and inelastic resonance scattering and the intensity of the stimulated emission
of the photons have been obtained.

Selective information transfer in spin ring networks by landscape shaping
control has the property that the error $1-\mathrm{prob}$, where
$\mathrm{prob}$ is the transfer success probability, and the sensitivity of the
probability to spin coupling errors are "positively correlated," meaning that
both are statistically increasing across a family of controllers of increasing
error. Here, we examine the rank correlation between the error and another
measure of performance-the logarithmic sensitivity-used in robust control to
formulate the fundamental limitations. Contrary to error versus sensitivity,
the error versus logarithmic sensitivity correlation is less obvious, because
its weaker trend is made difficult to detect by the noisy behavior of the
logarithmic sensitivity across controllers of increasing error numerically
optimized in a challenging landscape. This results in the Kendall $\tau$ test
for rank correlation between the error and the log sensitivity to be
pessimistic with poor confidence. Here it is shown that the Jonckheere-Terpstra
test, because it tests the Alternative Hypothesis of an ordering of the medians
of some groups of log sensitivity data, alleviates this problem and hence
singles out cases of anti-classical behavior of "positive correlation" between
the error and the logarithmic sensitivity.

Replacements for Fri, 9 Dec 16

Comments: 19 pages, 6 figures. Definition 6 is added; the statements of Theorem 1, Theorem 7, Lemma 11, Lemma 12, and Remark in the end of Section II are modified; and the proofs in Appendix B-C and B-D are modified. The main results are unchanged

Comments: 44 pages, significantly extended version, tight and close-to-tight continuity bounds for capacities of finite-dimensional (v.2) and infinite-dimensional (v.3) quantum channels are added, in v.3 the section devoted to the Holevo capacity is revised, in v.4 applications to the multi-mode quantum oscillator are revised and extended

Comments: 8 pages. Talk presented at the International Workshop "Strong Field Problems in Quantum Field Theory" (June 06-11, Tomsk, Russia), which complements arXiv:1604.03027, including extended historical background in Introduction, as well as additional discussions of the gauge-invariant Gribov horizon, and soft nilpotency; part with BRST invariant local Gribov-Zwanziger theory restored in (39), (40)